From Differential Equations to Multilayer Neural Network Models

  • Tatiana T. Kaverzneva
  • Galina F. MalykhinaEmail author
  • Dmitriy A. Tarkhov
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11554)


A method for constructing multilayer neural network approximations of solutions of differential equations, based on the finite difference method, is proposed. The advantage of the method is the possibility of obtaining a neural network model of arbitrarily high accuracy without a time-consuming learning procedure. The solution is given by an analytical expression, explicitly including the parameters of the problem. The resulting neural network can, if necessary, be retrained according to the usual algorithm. The method is illustrated by the example of solving a particular ordinary second-order differential equation.


The differential equation Multilayer approximate solution Neural network model Deformation Elastic thread 



The article was prepared on the basis of scientific research carried out with the financial support of the Russian Science Foundation grant (project No. 18-19-00474).


  1. 1.
    Haykin, S.: Neural Networks: A Comprehensive Foundation. Prentice Hall, Upper Saddle River (1999)Google Scholar
  2. 2.
    Lagaris, I.E., Likas, A., Fotiadis, D.I.: Artificial neural networks for solving ordinary and partial differential equations. IEEE Trans. Neural Netw. 9(5), 987–1000 (1998)Google Scholar
  3. 3.
    Dissanayake, M.W.M.G., Phan-Thien, N.: Neural-network-based approximations for solving partial differential equations. Commun. Numer. Methods Eng. 10(3), 195–201 (1994)Google Scholar
  4. 4.
    Fasshauer, G.E.: Solving differential equations with radial basis functions: multilevel methods and smoothing. Adv. Comput. Math. 11, 139–159 (1999)Google Scholar
  5. 5.
    Fornberg, B., Larsson, E.A.: Numerical study of some radial basis function based solution methods for elliptic PDEs. Comput. Math. Appl. 46, 891–902 (2003)Google Scholar
  6. 6.
    Galperin, E., Pan, Z., Zheng, Q.: Application of global optimization to implicit solution of partial differential equations. Comput. Math. Appl. 25(10/11), 119–124 (1993)Google Scholar
  7. 7.
    Galperin, E., Zheng, Q.: Solution and control of PDE via global optimization methods. Comput. Math. Appl. 25(10/11), 103–118 (1993)Google Scholar
  8. 8.
    Sharan, M., Kansa, E.J., Gupta, S.: Application of the multiquadric method to the numerical solution of elliptic partial differential equations. Appl. Math. Comput. 84, 275–302 (1997)Google Scholar
  9. 9.
    Vasilyev, A., Tarkhov, D., Guschin, G.: Neural networks method in pressure gauge modeling. In: Proceedings of the 10th IMEKO TC7 International Symposium on Advances of Measurement Science, Saint-Petersburg, Russia, vol. 2, pp. 275–279 (2004)Google Scholar
  10. 10.
    Tarkhov, D., Vasilyev, A.: New neural network technique to the numerical solution of mathematical physics problems. I: simple problems. Opt. Mem. Neural Netw. (Inf. Opt.) 14(1), 59–72 (2005)Google Scholar
  11. 11.
    Tarkhov, D., Vasilyev, A.: New neural network technique to the numerical solution of mathematical physics problems. II: complicated and nonstandard problems. Opt. Mem. Neural Netw. (Inf. Opt.) 14(2), 97–122 (2005)Google Scholar
  12. 12.
    Vasilyev, A., Tarkhov, D.: Mathematical models of complex systems on the basis of artificial neural networks. Nonlinear Phenom. Complex Syst. 17(2), 327–335 (2014)Google Scholar
  13. 13.
    Shemyakina, T.A., Tarkhov, D.A., Vasilyev, A.N.: Neural network technique for processes modeling in porous catalyst and chemical reactor. In: Cheng, L., Liu, Q., Ronzhin, A. (eds.) ISNN 2016. LNCS, vol. 9719, pp. 547–554. Springer, Cham (2016). Scholar
  14. 14.
    Gorbachenko, V.I., Lazovskaya, T.V., Tarkhov, D.A., Vasilyev, A.N., Zhukov, M.V.: Neural network technique in some inverse problems of mathematical physics. In: Cheng, L., Liu, Q., Ronzhin, A. (eds.) ISNN 2016. LNCS, vol. 9719, pp. 310–316. Springer, Cham (2016). Scholar
  15. 15.
    Budkina, E.M., Kuznetsov, E.B., Lazovskaya, T.V., Leonov, S.S., Tarkhov, D.A., Vasilyev, A.N.: Neural network technique in boundary value problems for ordinary differential equations. In: Cheng, L., Liu, Q., Ronzhin, A. (eds.) ISNN 2016. LNCS, vol. 9719, pp. 277–283. Springer, Cham (2016). Scholar
  16. 16.
    Lozhkina, O., Lozhkin, V., Nevmerzhitsky, N., Tarkhov, D., Vasilyev, A.: Motor transport related harmful PM2.5 and PM10: from on road measurements to the modelling of air pollution by neural network approach on street and urban level. J. Phys.: Conf. Ser. 772 (2016).–6596/772/1/012031
  17. 17.
    Kaverzneva, T., Lazovskaya, T., Tarkhov, D., Vasilyev, A.: Neural network modeling of air pollution in tunnels according to indirect measurements. J. Phys.: Conf. Ser. 772 (2016).
  18. 18.
    Lazovskaya, T.V., Tarkhov, D.A., Vasilyev, A.N.: Parametric neural network modeling in engineering. Recent Patents Eng. 11(1), 10–15 (2017)Google Scholar
  19. 19.
    Antonov, V., Tarkhov, D., Vasilyev, A.: Unified approach to constructing the neural network models of real objects. Part 1. Math. Models Methods Appl. Sci. 41(18), 9244–9251 (2018)Google Scholar
  20. 20.
    Lazovskaya, T., Tarkhov, D.: Multilayer neural network models based on grid methods. In: IOP Conference Series: Materials Science and Engineering, vol. 158 (2016).
  21. 21.
    Hairer, E., Norsett, S.P., Wanner, G.: Solving Ordinary Differential Equations I: Nonstiff Problem. Springer, Berlin (1987). Scholar
  22. 22.
    Bolgov, I., Kaverzneva, T., Kolesova, S., Lazovskaya, T., Stolyarov, O., Tarkhov, D.: Neural network model of rupture conditions for elastic material sample based on measurements at static loading under different strain rates. J. Phys.: Conf. Ser. 772 (2016).
  23. 23.
    Aranda-Iglesias, D., Vadillo, G., Rodríguez-Martínez, J.A., Volokh, K.Y.: Modeling deformation and failure of elastomers at high strain rates. Mech. Mater. 104, 85–92 (2017)Google Scholar
  24. 24.
    Zéhil, G.-P., Gavin, H.P.: Unified constitutive modeling of rubber-like materials under diverse loading conditions. Int. J. Eng. Sci. 62, 90–105 (2013)Google Scholar
  25. 25.
    Hearle, J.W.S.: One-Dimensional Textiles. Handbook of Technical Textiles. Elsevier, Amsterdam (2016)Google Scholar
  26. 26.
    McKenna, H.A., Hearle, J.W.S., O’Hear, N.: Handbook of Fibre Rope Technology. Handbook of Fibre Rope Technology. Elsevier, Amsterdam (2004)Google Scholar
  27. 27.
    Weller, S.D., Johanning, L., Davies, P., Banfield, S.J.: Synthetic mooring ropes for marine renewable energy applications. Renew. Energy 83, 1268–1278 (2015)Google Scholar
  28. 28.
    Vasilyev, A.N., Tarkhov, D.A., Tereshin, V.A., Berminova, M.S., Galyautdinova, A.R.: Semi-empirical neural network model of real thread sagging. In: Kryzhanovsky, B., Dunin-Barkowski, W., Redko, V. (eds.) NEUROINFORMATICS 2017. SCI, vol. 736, pp. 138–144. Springer, Cham (2018). Scholar
  29. 29.
    Zulkarnay, I.U., Kaverzneva, T.T., Tarkhov, D.A., Tereshin, V.A., Vinokhodov, T.V., Kapitsin, D.R.: A two-layer semi empirical model of nonlinear bending of the cantilevered beam. J. Phys.: Conf. Ser. 1044 (2018).
  30. 30.
    Bortkovskaya, M.R., et al.: Modeling of the membrane bending with multilayer semi-empirical models based on experimental data. In: Proceedings of the 2nd International Scientific Conference “Convergent Cognitive Information Technologies” (Convergent’2017) (2017).

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Tatiana T. Kaverzneva
    • 1
  • Galina F. Malykhina
    • 1
    • 2
    Email author
  • Dmitriy A. Tarkhov
    • 1
  1. 1.Peter the Great St. Petersburg Polytechnic UniversitySaint PetersburgRussia
  2. 2.Russian State Scientific Center for Robotics and Technical CyberneticsSaint PetersburgRussia

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